Counterexample to a Boesch's Conjecture

Abstract

A key issue in network reliability analysis. A graph with $n$ nodes and whose $e$ edges fail independently with probability $p$ is an \emph{Uniformly Most Reliable Graph} (UMRG) if it has the highest reliability among all graphs with the same order and size for every value of $p$. The \emph{all-terminal reliability} is a polynomial in $p$ which defines the probability of a network to remain connected if some of its components fail. If the coefficients of the reliability polynomial are maximized by a graph, that graph is called \textit{Strong Uniformly Most Reliable Graph} (SUMRG) and it should be UMRG. An exhaustive computer search of the SUMRG with vertices up to 9 is done. Regular graphs with 10 to 14 vertices that maximize tree number are proposed as candidates to UMRG. As an outstanding result a UMRG with 9 vertices and 18 edges which has girth 3 is found, so smaller than the conjectured by Boesch in 1986. A new conjecture about UMRG's topology is posed here: the $(n,e)$-UMRG is $\overline{(k-1)C_3\cup C_{3+r}}$ whenever $n=3k+r$,$n\geq5$ and $e={n(n-3)}/{2}$. A reformulation of Boesch's conjecture is presented stating that if a $(n, {kn}/{2})$-UMRG exists and it has girth $g$, then it has maximum girth among all $k$-regular $(n,{kn}/{2})$ graphs and minimum number of $g$-cycles among those $k$-regular $(n,{kn}/{2})$ graphs with girth $g$.